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Pascal’s triangle is filled with a wealth of interesting patterns, something I never learnt during secondary school when first introduced to this mathematical object. In this post, inspired by a Numberphile video, I want to show you some of these patterns that you might have never seen before.

Firstly, what is Pascal’s Triangle?

Pascal’s triangle is a triangular array of the binomial coefficients.

To construct Pascal’s triangle, start with the two top rows, which are 1 and 1 1. To find any number in the next row, add the two numbers above it.

At the beginning and end of each row, where there is only one number above, write 1.

Symmetric

Pascal’s triangle is symmetric, a fact which follows directly from the formula for the binomial coefficient:

It can also be seen by how the triangle is constructed, i.e. “by the interpretation of the entries as the number of ways to get from the top to a given spot in the triangle“.

Sum of entries in row n equals 2n

This fact can be proved by induction. The main point of the argument is that each entry in row n is added to two entries below. This gives us:

Hence, the sum of entries in row n+1 is twice the sum of entries in row n.

Hockey Stick Pattern

In Pascal’s words: “In every arithmetical triangle each cell is equal to the sum of all the cells of the preceding row from its column to the first, inclusive”. This is easier to explain using a diagram:

Mathematically, this can be proved by induction and is denoted in the following way:

Fibonacci Numbers

To see this, let us first rearrange Pascal’s triangle.

The successive Fibonacci numbers are the sums of the entries on the marked diagonals:

1 = 1

1 = 1

2 = 1 + 1

3 = 1 + 2

5 = 1 + 3 + 1

8 = 1 + 4 + 3

13 = 1 + 5 + 6 + 1

etc.

e

Recently, the Harlan brothers highlighted that e is hidden in Pascal’s triangle. This was discovered by considering products rather than sums.

Denoting Sn as the product of the terms in the nth row, as n tends to infinity, we find that:

The Sierpinski Triangle, or Sierpinski Sieve, is a fractal described by Polish Mathematician Sierpinski in 1915, although it appeared in Italian art from the 13th century. It has an overall shape of an equilateral triangle, and is subdivided recursively into smaller equilateral triangles.

Source: Wolfram Mathworld

Constructing a Sierpinski Triangle

STEP 1:

Start with an equilateral triangle.

STEP 2:

Connect the midpoints of each side, hence dividing it into 4 smaller congruent equilateral triangles.

STEP 3:

Now cut out the triangle in the centre.

STEP 4:

Repeat steps 2 and 3 with each of the remaining smaller triangles.

Properties

If we let be the number of black triangles after iteration n, be the length of a side of a triangle, and be the fractional area which is black after the nth iteration, then: